# Filling Gaps in Daily Precipitation Series Using Regression and Machine Learning in Inter-Andean Watersheds

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}(2250, 1167 and 6079 km

^{2}, MOC basins, respectively). The area is characterized by a significant latitudinal gradient that goes from 0 to 6500 masl; above 2500 masl is the wet watershed area [32]. The rivers flow from east to west from the Andes to the Pacific Ocean, with bare, steep slopes that favor significant swelling, floods, and erosion during heavy rainfall episodes [9]. In addition, in normal conditions, the region is influenced by the South Pacific High, in combination with the Humboldt current that produces dry, stable conditions with moist air trapped below the inversion layer at about 1000 masl, and it presents major seasonal and interannual precipitation variability [9,11,13].

#### 2.2. Methods

#### 2.2.1. Collection of Available Information

#### 2.2.2. Exploratory Data Analysis (EDA)

_{i}to denote a specific value (for example, cumulative annual precipitation or mean annual temperature) in the year (or other unit of time) i. In addition, X

_{j}denotes one of the surrounding reference sites (jth of a total of k) and X

_{ji}a specific value from that site. The following equations were used to detect the relative non-homogeneities (traditionally used in precipitation studies):

_{i}is the ratio in Equation (1) and the difference in Equation (2) in a specific year i; Ŷ represents the multi-annual mean of the candidate time series; and ρ

_{j}is the correlation coefficient between the test variable Y and the reference variable X

_{j}[36,38,39]. This method is implemented in the Climatol package for R language [34]. Climatol has three normalization methods: division by mean values, subtraction of means, and complete standardization; here, we opted for subtraction of means, as the minimum precipitation values can be zero [34,40,41]. On a preliminary basis, Climatol was run for a monthly time step, identifying breaks; based on these breaks, Climatol was run again for a daily time step. The results show graphs of absolute maximum autocorrelation (ACmx), SNHT, root mean square error (RMSE), and percentage of original data (POD).

#### 2.2.3. Regionalization Process

_{k}represents the number of attribute vectors in group k; y

^{k}

_{ij}denotes the rescaled value of attribute j in attribute vector i assigned to group k; and y

^{k}

_{•j}is the mean value of attribute j for group k (Equation (4)) [43,45].

_{i}and extended mean precipitation P

_{j}, which is achieved by minimizing the expression [10,11,45]:

_{ij}is annual precipitation at station j in year i; P

_{j}is mean precipitation extended to a period of N years; and, finally, Z

_{i}is the regional rainfall index of year i. This process was carried out using the Hydracces program [48].

#### 2.2.4. Gap-Filling Model

_{m}) for MRM. LRM is a computing procedure based on the alternate least squares algorithm (ALS) [49]. It has two steps: first estimating the relationship between predictors and missing values and then using the trend equation to fill the gaps [50], in accordance with Equation (6):

_{i}is the dependent variable; X

_{1}, X

_{2},…X

_{m}are the independent variables; a is the intersection; b

_{1}, b

_{2},…b

_{m}are the multiple regression coefficients, estimated using the method of least squares; and C is the error term [50,51]. ML is a scientific discipline in the artificial intelligence field that creates systems that learn automatically [8,14]. For gap filling using this technique, the data available at each station were divided randomly to generate a training dataset (train) and test dataset (test) in proportions of 75% and 25%, respectively [8]. The algorithms implemented were MRM, K-nearest neighbors (KNN), gradient boosting trees (GBT), and random forest (RF). In addition, an optimization process was carried out, generating OML-MRM, OML-KNN, OML-GBT, and OML-RF models. These algorithms were implemented using the Python programming language. KNN is a non-parametric method that can be used for both classification and regression.

#### 2.2.5. Bayesian Optimization

#### 2.2.6. Evaluation Metrics

^{2}), root mean square error (RMSE), Nash–Sutcliffe coefficient (NSE) and percentage bias (PBIAS) were used [8,51,58]. All of them are mathematically expressed as Equations (10)–(13), respectively:

## 3. Results

#### 3.1. Analysis of Missing Data, Outliers, and Homogenization

#### 3.2. Regionalization Analysis

#### 3.3. Analysis of the Series Gap-Filling Process

_{m}(see Table 5).

_{m}stations corresponding to each target station. The Y and X

_{m}stations for each homogenous region are shown in Table 5.

_{m}variables, ML was first applied for default parameter values using the ML-MRM, ML-KNN, ML-GBT, and ML-RF models. It was also applied using parameters called hyperparameters, generating the OML-MRM, OML-KNN, OML-GBT, and OML-RF models. This process allowed the model parameters to be optimized. The parameter and hyperparameter values used in the algorithms created in Python are shown in Table 6.

#### 3.4. Assessment of Model Performance

^{2}, RMSE, NSE and PBIAS—were used for both datasets (training and test). The obtained results are presented in Table 7 and Tables S6 and S7 (Supplementary Material). These statistics were calculated for the 2001–2019 period; periods with missing data were not considered.

^{2}values for the dataset (training and test) present a correlation between the Y and X variables in each model. For the Ayaviri station, which belongs to homogenous region 1 (see Table 7), the ML-RF model gives the best R

^{2}value (0.89) for the training data; however, for the test dataset, this value is reduced by nearly half (R

^{2}= 0.45). For optimized ML, the training and test R2 values are close to each other, and in some cases, the R

^{2}values are better for the test datasets than the training datasets. RMSE is a measure of the variance of residuals, which allows the magnitude of deviation of simulated values from observed values to be quantified; the LRM model presents the greatest RMSE (3.15) for the Ayaviri station. It was also observed that the test dataset generally presents a lower RMSE, particularly with the optimized ML models (OML-GBT and OML-RF).

## 4. Discussion

^{2}= 0.89, RMSE = 1.36, NSE = 0.88, and PBIAS = −1.73). Figure 11b shows the Taylor diagram for the Ayaviri station for the test dataset; the OML-GBT and OML-RF present the best results (R

^{2}= 0.71, RMSE = 2.05, NSE = 0.71, and PBIAS = 0.00 and R

^{2}= 0.70, RMSE = 2.14, NSE = 0.68 y PBIAS = 1.01, respectively). The analyses of the other stations (Huarochiri, San Lazaro de Escomarca, and Langa), are shown in Figure 11c–h; all these stations are located in homogenous region 1, and the values of the results obtained for them are similar to those of the Ayaviri station. Likewise, it is observed that in terms of the statistical metrics for the training and test datasets, the optimized ML models present the best results, particularly the OML-GBT and OML-RF models. The results of the analysis of the statistical metrics are shown in the figures. For the Ayaviri station, the OML-RF model presents a slight underestimation, while the results of the OML-GBT model are more efficient. Finally, in regions 2 and 4, Figures S3 and S4 respectively (Supplementary Material), the OML-GBT and OML-RF present the best results in terms of statistical metrics.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Elevation of the study area, main rivers, selected watershed boundaries, and location of rainfall stations.

**Figure 3.**Missing daily precipitation data: quantity of unavailable daily precipitation data as a percentage by station.

**Figure 5.**Correlogram between stations: (

**a**) Daily precipitation series and (

**b**) Monthly precipitation series.

**Figure 6.**Homogeneity analysis statistics: (

**a**) Station maximum absolute autocorrelation (ACmx), (

**b**) Standard normal homogeneity test (SNHT), (

**c**) Root mean squared error (RMSE) and (

**d**) Percentage of original data (POD).

**Figure 11.**Taylor diagrams that show a statistical comparison (normalized standard deviation and correlation coefficient) of observed precipitation and modeled precipitation based on precipitation datasets (training and test) for four stations: (

**a**) Ayaviri (training), (

**b**) Ayaviri (test), (

**c**) Huarochiri (training), (

**d**) Huarochiri (test), (

**e**) San Lazaro de Escomarca (training), (

**f**) San Lazaro de Escomarca (test), (

**g**) Langa (training), and (

**h**) Langa (test).

ID | Stations | Coordinates | Altitude | Observed Data | Missing Data | |||
---|---|---|---|---|---|---|---|---|

Latitude | Longitude | (masl) | No of Data | (%) | No of Data | (%) | ||

1 | Ayaviri | −12.38 | −76.13 | 3228 | 6881 | 99.2 | 58 | 0.8 |

2 | Cañete | −13.07 | −76.32 | 158 | 3830 | 55.2 | 3109 | 44.8 |

3 | Carania | −12.34 | −75.87 | 3875 | 6939 | 100 | 0 | 0 |

4 | Huancata | −12.22 | −76.22 | 2700 | 6939 | 100 | 0 | 0 |

5 | Huangascar | −12.9 | −75.83 | 2533 | 6908 | 99.6 | 31 | 0.4 |

6 | Huañec | −12.29 | −76.14 | 3205 | 6939 | 100 | 0 | 0 |

7 | Huarochiri | −12.13 | −76.23 | 3154 | 6787 | 97.8 | 152 | 2.2 |

8 | Langa | −12.13 | −76.42 | 2863 | 6484 | 93.4 | 455 | 6.6 |

9 | Pacaran | −12.83 | −76.07 | 700 | 5132 | 74 | 1807 | 26 |

10 | San Juan de Yanac | −13.21 | −75.79 | 2550 | 6482 | 93.4 | 457 | 6.6 |

11 | San Lazaro de Escomarca | −12.18 | −76.35 | 3758 | 6486 | 93.5 | 453 | 6.5 |

12 | San Pedro de Pilas | −12.45 | −76.22 | 2600 | 6909 | 99.6 | 30 | 0.4 |

13 | Socsi | −13.03 | −76.19 | 500 | 4687 | 67.5 | 2252 | 32.5 |

14 | Tanta | −12.12 | −76.02 | 4323 | 6819 | 98.3 | 120 | 1.7 |

15 | Vilca | −12.11 | −75.83 | 3864 | 6297 | 90.7 | 642 | 9.3 |

16 | Yauricocha | −12.32 | −75.72 | 4675 | 6818 | 98.3 | 121 | 1.7 |

17 | Yauyos | −12.49 | −75.91 | 2327 | 6878 | 99.1 | 61 | 0.9 |

Stations | ACmx | SNHT | RMSE | POD |
---|---|---|---|---|

Ayaviri | 0.19 | 47.4 | 2.9 | 99 |

Cañete | 0.65 | 228.0 | 1.3 | 55 |

Carania | 0.20 | 26.1 | 2.6 | 100 |

Huancata | 0.33 | 95.0 | 2.4 | 100 |

Huangascar | 0.14 | 35.9 | 1.9 | 99 |

Huañec | 0.29 | 68.7 | 2.2 | 100 |

Huarochiri | 0.13 | 55.0 | 2.7 | 97 |

Langa | 0.08 | 80.9 | 2.0 | 93 |

Pacaran | 0.73 | 166.1 | 1.3 | 73 |

San Juan de Yanac | 0.10 | 21.5 | 1.5 | 93 |

San Lazaro de Escomarca | 0.32 | 20.9 | 3.4 | 93 |

San Pedro de Pilas | 0.15 | 13.4 | 2.0 | 99 |

Socsi | 0.78 | 40.8 | 1.4 | 67 |

Tanta | 0.34 | 155.6 | 4.8 | 98 |

Vilca | 0.34 | 30.5 | 3.9 | 90 |

Yauricocha | 0.36 | 38.6 | 4.6 | 98 |

Yauyos | 0.08 | 9.1 | 1.9 | 99 |

Station | Time | Standard | Station/Vector |
---|---|---|---|

(Years) | Deviation | Correlation | |

Langa | 16 | 0.252 | 0.882 |

San Lazaro de Escomarca | 16 | 0.263 | 0.664 |

Ayaviri | 17 | 0.116 | 0.904 |

Huancata | 19 | 0.341 | 0.863 |

Huañec | 19 | 0.187 | 0.751 |

Huarochiri | 14 | 0.159 | 0.851 |

Carania | 19 | 0.191 | 0.679 |

Ayaviri | 1 | ||||||

Carania | 0.48 | 1 | |||||

Huancata | 0.60 | 0.47 | 1 | ||||

Huañec | 0.51 | 0.43 | 0.49 | 1 | |||

Huarochiri | 0.56 | 0.55 | 0.58 | 0.46 | 1 | ||

San Lazaro de Escomarca | 0.45 | 0.40 | 0.43 | 0.39 | 0.44 | 1 | |

Langa | 0.45 | 0.38 | 0.46 | 0.38 | 0.46 | 0.55 | 1 |

Ayaviri | Carania | Huancata | Huañec | Huarochiri | San Lazaro de Escomarca | Langa |

Regions | Target Station (Y) | Predictor Station (X) | Multiple Predictor Stations (X_{m}) |
---|---|---|---|

Region 1 | Ayaviri | Huancata | Huancata, Langa, San Lazaro de Escomarca, Huañec, Huarochiri, Carania |

Huarochiri | Huancata | Huancata, Langa, San Lazaro de Escomarca, Ayaviri, Huañec, Carania | |

San Lazaro de Escomarca | Langa | Langa, Ayaviri, Huancata, Huañec, Huarochiri, Carania | |

Langa | San Lazaro de Escomarca | San Lazaro de Escomarca, Ayaviri, Huancata, Huañec, Huarochiri, Carania | |

Region 2 | San Pedro de Pilas | Huangascar | Huangascar, San Juan de Yanac, Yauyos |

Huangascar | San Pedro de Pilas | San Pedro de Pilas, Yauyos, San Juan de Yanac | |

Yauyos | San Pedro de Pilas | San Pedro de Pilas, Huangascar, San Juan de Yanac | |

San Juan de Yanac | San Pedro de Pilas | San Pedro de Pilas, Huangascar, Yauyos | |

Region 4 | Tanta | Vilca | Vilca and Yauricocha |

Yauricocha | Vilca | Vilca and Tanta | |

Vilca | Yauricocha | Yauricocha and Tanta |

Algorithm | Parameters [Values] | Hyperparameters [Values] |
---|---|---|

Multiple Regression | alpha [1] | alpha [logspace(–5, 5, 500)] |

solver [‘auto’] | solver [‘auto’] | |

modelo[Ridge] | modelo[Ridge] | |

K-nearest neighbors | n_neighbors [5] | n_neighbours [linspace(1, 100, 500] |

leaf_size [30] | leaf_size [1, 3] | |

algoritm [‘auto’] | algoritm [‘auto’] | |

modelo[KNeighborsRegressor] | modelo[KNeighborsRegressor] | |

Gradient boosting tree | n_estimators [100] | n_estimators [50, 100, 1000, 2000] |

max_feature [‘none’] | max_feature [‘auto’, 3, 5, 7] | |

max_depth [3] | max_depth [‘None’, 3, 5, 10, 20] | |

subsample [1] | subsample [0.5, 0.7, 1] | |

modelo[GradientBoostingRegressor] | modelo[GradientBoostingRegressor] | |

Random forest | n_estimators [100] | n_estimators [50, 100, 1000, 2000] |

max_feature [‘auto’] | max_feature [‘auto’, 3, 5, 7] | |

max_depth [‘None’] | max_depth [‘None’, 3, 5, 10, 20] | |

modelo[RandomForestRegressor] | modelo[RandomForestRegressor] |

Stations | Samples | Statistics | LRM | MRM | Machine Learning | Optimized Machine Learning | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MRM | KNN | GBT | RF | MRM | KNN | GBT | RF | |||||

Ayaviri | Train | R^{2} | 0.36 | 0.49 | 0.48 | 0.57 | 0.64 | 0.89 | 0.48 | 0.49 | 0.59 | 0.59 |

Train | RMSE | 3.15 | 2.81 | 2.87 | 2.61 | 2.39 | 1.36 | 2.87 | 2.89 | 2.55 | 2.58 | |

Train | NSE | 0.36 | 0.49 | 0.48 | 0.57 | 0.64 | 0.88 | 0.48 | 0.47 | 0.59 | 0.58 | |

Train | PBIAS | 0.00 | 0.00 | 0.00 | 3.92 | 0.00 | −1.73 | 0.00 | 0.45 | 0.00 | 0.38 | |

Test | R^{2} | 0.52 | 0.38 | 0.49 | 0.45 | 0.52 | 0.48 | 0.71 | 0.70 | |||

Test | RMSE | 2.62 | 3.03 | 2.75 | 2.86 | 2.62 | 2.83 | 2.05 | 2.14 | |||

Test | NSE | 0.52 | 0.36 | 0.47 | 0.43 | 0.52 | 0.44 | 0.71 | 0.68 | |||

Test | PBIAS | 0.00 | 0.67 | −8.46 | −10.65 | 0.00 | 21.64 | 0.00 | 1.01 | |||

Huarochiri | Train | R^{2} | 0.34 | 0.49 | 0.49 | 0.60 | 0.65 | 0.92 | 0.49 | 0.51 | 0.60 | 0.61 |

Train | RMSE | 3.12 | 2.74 | 2.80 | 2.47 | 2.32 | 1.19 | 2.80 | 2.76 | 2.49 | 2.48 | |

Train | NSE | 0.34 | 0.49 | 0.49 | 0.60 | 0.65 | 0.91 | 0.49 | 0.50 | 0.60 | 0.60 | |

Train | PBIAS | 0.00 | 0.00 | 0.00 | 6.48 | 0.00 | −1.39 | 0.00 | 4.38 | 0.00 | 0.47 | |

Test | R^{2} | 0.52 | 0.41 | 0.51 | 0.49 | 0.53 | 0.53 | 0.73 | 0.73 | |||

Test | RMSE | 2.54 | 2.83 | 2.58 | 2.64 | 2.51 | 2.57 | 1.93 | 1.96 | |||

Test | NSE | 0.52 | 0.40 | 0.50 | 0.48 | 0.53 | 0.51 | 0.72 | 0.71 | |||

Test | PBIAS | −1.72 | 7.12 | −5.63 | −9.30 | 0.00 | 18.36 | 0.00 | 0.95 | |||

San Lazaro de Escomarca | Train | R^{2} | 0.30 | 0.38 | 0.38 | 0.49 | 0.65 | 0.90 | 0.38 | 0.41 | 0.54 | 0.45 |

Train | RMSE | 3.44 | 3.22 | 3.16 | 2.87 | 2.42 | 1.41 | 3.17 | 3.11 | 2.75 | 3.03 | |

Train | NSE | 0.30 | 0.38 | 0.38 | 0.49 | 0.64 | 0.88 | 0.38 | 0.40 | 0.53 | 0.43 | |

Train | PBIAS | 0.00 | 0.00 | 0.00 | 7.01 | 0.00 | −1.96 | 0.00 | 10.88 | 0.00 | 0.14 | |

Test | R^{2} | 0.42 | 0.28 | 0.34 | 0.37 | 0.41 | 0.43 | 0.73 | 0.56 | |||

Test | RMSE | 3.33 | 3.73 | 3.55 | 3.46 | 3.34 | 3.35 | 2.31 | 2.98 | |||

Test | NSE | 0.42 | 0.27 | 0.33 | 0.37 | 0.41 | 0.41 | 0.72 | 0.53 | |||

Test | PBIAS | 0.00 | 16.25 | 9.41 | 1.78 | 0.00 | 14.86 | 0.00 | −0.05 | |||

Langa | Train | R^{2} | 0.30 | 0.39 | 0.40 | 0.53 | 0.68 | 0.93 | 0.40 | 0.45 | 0.59 | 0.60 |

Train | RMSE | 1.98 | 1.85 | 1.85 | 1.64 | 1.37 | 0.71 | 1.85 | 1.80 | 1.55 | 1.55 | |

Train | NSE | 0.30 | 0.39 | 0.40 | 0.53 | 0.67 | 0.91 | 0.40 | 0.43 | 0.58 | 0.58 | |

Train | PBIAS | 0.00 | 0.00 | 0.00 | 5.79 | 0.00 | −3.09 | 0.00 | 10.61 | 0.00 | 0.60 | |

Test | R^{2} | 0.36 | 0.24 | 0.32 | 0.31 | 0.37 | 0.36 | 0.70 | 0.70 | |||

Test | RMSE | 1.85 | 2.09 | 1.94 | 1.98 | 1.83 | 1.87 | 1.28 | 1.33 | |||

Test | NSE | 0.35 | 0.17 | 0.28 | 0.26 | 0.37 | 0.34 | 0.69 | 0.67 | |||

Test | PBIAS | −4.32 | 0.76 | −7.22 | −16.36 | 0.00 | 17.93 | 0.00 | 1.23 |

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## Share and Cite

**MDPI and ACS Style**

Portuguez-Maurtua, M.; Arumi, J.L.; Lagos, O.; Stehr, A.; Montalvo Arquiñigo, N.
Filling Gaps in Daily Precipitation Series Using Regression and Machine Learning in Inter-Andean Watersheds. *Water* **2022**, *14*, 1799.
https://doi.org/10.3390/w14111799

**AMA Style**

Portuguez-Maurtua M, Arumi JL, Lagos O, Stehr A, Montalvo Arquiñigo N.
Filling Gaps in Daily Precipitation Series Using Regression and Machine Learning in Inter-Andean Watersheds. *Water*. 2022; 14(11):1799.
https://doi.org/10.3390/w14111799

**Chicago/Turabian Style**

Portuguez-Maurtua, Marcelo, José Luis Arumi, Octavio Lagos, Alejandra Stehr, and Nestor Montalvo Arquiñigo.
2022. "Filling Gaps in Daily Precipitation Series Using Regression and Machine Learning in Inter-Andean Watersheds" *Water* 14, no. 11: 1799.
https://doi.org/10.3390/w14111799